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Theorem rmobida 2798
 Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
rmobida.1 xφ
rmobida.2 ((φ x A) → (ψχ))
Assertion
Ref Expression
rmobida (φ → (∃*x A ψ∃*x A χ))

Proof of Theorem rmobida
StepHypRef Expression
1 rmobida.1 . . 3 xφ
2 rmobida.2 . . . 4 ((φ x A) → (ψχ))
32pm5.32da 622 . . 3 (φ → ((x A ψ) ↔ (x A χ)))
41, 3mobid 2238 . 2 (φ → (∃*x(x A ψ) ↔ ∃*x(x A χ)))
5 df-rmo 2622 . 2 (∃*x A ψ∃*x(x A ψ))
6 df-rmo 2622 . 2 (∃*x A χ∃*x(x A χ))
74, 5, 63bitr4g 279 1 (φ → (∃*x A ψ∃*x A χ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   ∈ wcel 1710  ∃*wmo 2205  ∃*wrmo 2617 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-eu 2208  df-mo 2209  df-rmo 2622 This theorem is referenced by:  rmobidva  2799
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